We know that line integrals of conservative vector fields are independent of path . let F be the field and r(t) be the paramatrization of the curve, if $\int_CF dr=0$ for every closed path in D, then F is conservative.
For the first image, a closed circle at center is an obvious counterexample, so the field is not conservative.
For the second image, it is conservative. However, I found that it is difficult to see that there's no conunterexample. For instance, if i pick a circle in first quadrant, how do you tell that $\int_CF dr=0$ for sure?
For such questions, it is obvious that no analytical answer is expected.
Your answers are correct, I think this is the type of reasoning that is expected.
Another approach would be as follows. Recall that the curl of a field is related to a rotation: more precisely, a rotation around the axis of the the curl, at a speed proportional to the magnitude of the curl. In the first image, we can see that the field tends to rotate counterclockwise, so the curl is not $\vec{0}$, which suggests that the field is not conservative. And for the seconde one, there seems to be no specific rotation, which implies that the curl is $\vec{0}$, and the field conservative.
This being said, we cannot be 100% sure, as you said it is difficult to see that there is no counterexample, or that there is no rotation whatsoever.
So to conclude, I would say keep things simple, I don't believe these exercises are meant to trick you, their purpose is to make you understand properties of vector fields from a graphical point of view.